Berezin integration in a Grassmann algebra is defined such that its algebraic properties are analogous to definite integration of ordinary functions: linearity (taking anticommutativity into account), scale invariance and independence from the integration variable. Up to normalization this means that
$$\int f(\theta) d\theta = \int (f_0 + f_1\theta) d\theta = f_1$$
I already have a good geometric picture of Grassmann numbers themselves as elements of an exterior algebra, but I've never understood the geometric meaning of Berezin integration. What exactly are we "summing" and over what domain? Why the strange scaling property $d(a\theta) = a^{-1} d\theta$?